You are riding a bicycle at a speed of 15 m/s. The bycicle has tires with diameters measuring 630 mm. You can approximate each tire as a solid cyllinder (to calculate inertia) with a mass of 0.5 kg. What is the total rotational kinetic energy of the bicycle?

To calculate the total rotational kinetic energy of the bicycle, we need to find the rotational kinetic energy of each tire and add them together.

The formula for rotational kinetic energy is:

KE_rot = (1/2) * I * ω^2

where KE_rot is the rotational kinetic energy, I is the moment of inertia, and ω is the angular velocity.

To find the moment of inertia for a solid cylinder, we use the formula:

I = (1/2) * m * r^2

where m is the mass of the object and r is the radius.

Let's start by finding the moment of inertia of each tire.

Given:
Mass of each tire, m = 0.5 kg
Diameter of each tire, d = 630 mm

First, we need to convert the diameter to radius:
Radius, r = d/2 = 630 mm / 2 = 315 mm = 0.315 m

Now, let's calculate the moment of inertia for each tire:
I = (1/2) * m * r^2 = (1/2) * 0.5 kg * (0.315 m)^2 = 0.0248 kg.m^2 (rounded to four decimal places)

Since we have two identical tires, we need to calculate the rotational kinetic energy for each tire and double it to get the total rotational kinetic energy.

Next, we need to calculate the angular velocity, ω, using the linear velocity, v.

Given:
Linear velocity, v = 15 m/s

To find the angular velocity, we need to use the formula:

v = ω * r

Solving for ω, we get:

ω = v / r = 15 m/s / 0.315 m = 47.619 rad/s (rounded to three decimal places)

Now, let's calculate the rotational kinetic energy for each tire:

KE_rot = (1/2) * I * ω^2 = (1/2) * 0.0248 kg.m^2 * (47.619 rad/s)^2 = 27.95 J (rounded to two decimal places)

Since we have two identical tires, the total rotational kinetic energy of the bicycle is:

Total KE_rot = 2 * KE_rot = 2 * 27.95 J = 55.90 J (rounded to two decimal places)

Therefore, the total rotational kinetic energy of the bicycle is 55.90 Joules.